# Thread: Determining probability of 6 variables based on an overall mean score of a game

1. ## Determining probability of 6 variables based on an overall mean score of a game

Hi all,

Sorry if this question is a little wordy, but bear with me...

If there is a game where a player can score either 60,20,15,5,3 or 1 points, and we know their average to a very accurate figure, say 39.219771. There must be a unique distribution of the possible scores to give this average. I.e., we could determine how often a player scores 60 points, how often they score 20 points, and so on.

This in effect would be 39.219771 = 60*x + 20*y + 15*z + 5*p + 3*q + 1*r, where x,y,z,p,q,r are probabilities between 0 and 1.

If the average was exactly 40.000000, then there may be lots of different possibilities that would produce the result. But because of the number of decimal places, 39.219771 for example, would have a unique distribution? Or atleast only a few possibilites?

I am trying to solve these variables for any average that I wanted. I.e. I could put in 41.645712 and I could solve for x,y,z,p,q,r.

I have mostly been trying to do this on excel and have been playing around with a few things, trying linear regression etc, but then I was wondering, would it require simulating all the variables until our number is produced? (simulating I am not too clued up on).

Any help greatly appreciated, let me know if any more info is required, I'm sure this is possible though?!

Thanks

2. ## Re: Determining probability of 6 variables based on an overall mean score of a game

You have a single equation in 6 unknowns. While the fact that they must all lie between 0 and 1 will restrict the solution space somewhat, there will still be an infinite number of solutions. The fact that 39.219771 or 41.645712 are "very accurate" or "have many decimal places" is irrelevant. (Actually 40.000000 has just as many decimal places and is just as accurate.)

3. ## Re: Determining probability of 6 variables based on an overall mean score of a game

I see what you are saying, and I agree to an extent. Then how would we go about:

a) solving to get a single solution (even if there are infinite).
b) We do have more than one linear equation, as x + y + z + p + q + r = 1. We could probably introduce a few more equations/caveats, to make the question more determinate? E.g., we could say x,y,z,p,q,r > 0.01 and x,y,z,p,q,r < 0.99.